Adaptability of Nonlinear
Equilibrium Models to Central
Planning
Zalai, E.
IIASA Collaborative Paper
November 1982
Zalai, E. (1982) Adaptability of Nonlinear Equilibrium Models to Central Planning. IIASA Collaborative Paper.
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ADAPTABILITY OF NONLINEAR
EQUILIBRIUM MODELS TO CENTRAL
PLANNING
Ern6 Zalai
November, 1982
CP-82-82
Presented at the Scientific Session
on Improving National Planning
organized by the Hungarian Planning
Office and Academy of Sciences,
Budapest, Septerber 21-22, 1982.
CoZZaborative Papers report work which has not been
performed solely at the International Institute for
Applied Systems Analysis and which has received only
limited review. Views or opinions expressed herein
do not necessarily represent those of the Institute,
its National Member Organizations, or other organizations supporting the work.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS
A-2361 Laxenburg, Austria
ABSTRACT
Linear multisectoral models have for long been applied in
the Hungarian national economic planning. Price-quantity correspondences and interaction, however, cannot easily be taken into
account in the traditional linear framework. Computable general
equilibrium modelers in the West have developed techniques which
use extensively price-quantity interdependences. However, since
they are usually presented with the controversial strict neoclassical interpretation, the possibility of their adaptation to
socialist planning models has been concealed. This paper reflects
on some results of a research investigating the possible adaptation of equilibrium modeling techniques to central planning
mode Is.
ADAPTABILITY OF NONLINEAR EQUILIBRIUM MODELS
TO CENTPAL PLANNING *
Ern6 Zalai
International Institute for Applied Systems Analysis
A-2361 Laxenburg, Austria
Institute for National Economic Planning
Karl M a m University of Economics, Budapest, Hungary
1.
INTRODUCTION
Linear multisectoral input-output and programming models
have becomemore or less integrated into the complex process of
planning in many socialist (centrally planned) economies.
These models concentrate on the production and use of economic
resources and commodities at some level of aggregation. Similar models are also used in both western and developing countries,
the differences in the economic environment and data sources
being reflected in the specification and purpose of the models.
The use of linear models has been paralleled by the development
of more complex, nonlinear models, most of which come under the
general heading of computable or applied general equilibrium
models.
The basic ideas of a multisectoral general equilibrium
growth model were first suggested by Johansen [I01 in 1959,
although full-scale implementation of large, nonlinear models
has become computationally feasible only lately.
*
This paper isbased on a research initiated by the author.
This research is a combined effort of experts in the Hungarian
Planning Office, Karl Marx University of Economics (Budapest)
and the International Institute for Applied SystemsAnalysis
(Laxenburg, Austria). References 5, 13 and 16 contain a more
detailed discussion of most of the issues only touched upon here.
Recent applications are described in references 1, 6, 7
and 8; models of this type developed at the International
Institute for Applied Systems Analysis (IIASA) are discussed in
references 3, 4, 1 1 and 16. Some of these models have been
designed to capture the interrelationships between economic,
spatial, and demographic processes.
The structure of general equilibrium models, the estimation procedures applied, and the theoretical explanations
associated with them generally follow the neoclassical tradition quite closely. The neoclassical approach has often been
criticized and even rejected in both the East and West, and
this partly explains the apparent lack of interest of central
planriing modelers in these models. It is not at all obvious
whether the models, or some of the techniques of applied general
equilibrium modeling, could be adapted for central planning
processes.
. .
The main purpose of this paper therefore is to highlight
the possibility, andexpected benefits, of incorporating nonlinear
multisectoral models of the general equilibrium type into the
planning methodology of socialist (centrally planned) economies.
First we will briefly review the major characteristics of
applied general equilibrium analysis and compare them with
those of a typical planning model. Next, with the help of a
simple illustrative model focusing on foreign trade relationships, we will provide some examples of why and how techniques
of general equilibrium models can be adopted for central planning. Finally, the possible advantages of such adaption are
considered in a somewhat wider context.
2.
GENERAL EQUILIBRIUM VERSUS OPTIMAL PLANNING MODELS
General competitive equilibrium theory* provides an abstract
partial model of the economic systems centered around the law
of supply and demand, and rational economic behavior. The
*
We will confine our attention to competitive or Walrasian
general equilibrium models, which provide a basis for less
classical equilibrium models and various disequilibrium models.
abstract economic theory of general equilibrium takes many important elements of the economy as d a t a and sets out to define
and determine the equilibrium within this postulated environment
in which only prices control economic decisions.
A p p l i e d general equilibrium models adopt a r e l a t i v e point
of view and try to estimate the likely consequences of various
changes in the economic environment by comparing the "base
equilibrium solution" with the solutions computed on the basis
of these changes. A typical approach may be summarized as follows.
A formal model of the necessary and sufficient conditions for
general equilibrium is developed. The observed state of the
economy is considered to be in equilibrium (base solution), and
many of the parameters of the model are statistically estimated
on the basis of this assumption. Next, by classifying the economic variables as endogenous or exogenous, the impact of assumed
changes in the exogenous variables is analyzed in terms of the
models solution. Thus, the equilibrium framework is used to
evaluate, consistently and in quatitative terms, the direction
of change of certain crucial interdependent economic variables.
A distinctive feature of general equilibrium analysis is that
it takes into account simultaneously real, price (cost) and
financial variables and their interaction.
The use and philosophy of macroeconomic planning models
could be summarized in the following way. Suppose that at some
stage in the planning process the coordinating unit decides to
summarize the calculations made so far, and as a result some
provisional values of the sectoral outputs, inputs, consumption,
etc., are therefore made available. The coordinating unit
wishes to know whether these more or less seperately planned
figures represent a consistent and balanced picture, and if not,
how this could be rectified. The unit also wishes to check how
certain changes in one part of the plan would affect other parts
of the provisional plan and its efficiency. In Hungary, formal
models support the process of checking the consistency, sensibility and efficiency of a draft plan (coordination process).
(See for example, reference 2 regarding current planning modeling
practice in Hungary.)
An economy-wide planning model, built into and upon the
traditional planning methodology of a socialist country, would
differ from the outlined general equilibrium models in several
respects. First, it would almost exclusively contain "real"
variables and relations reflecting physical constraints on
allocation. Second, because the prices used in a planning model
are either constant or planned, being predicted more or less
independently from "real" processes, the interdependence of
the real and value (prices, taxes, rate of return requirements,
etc.) variables would not be considered explicitly in the model.
Third, most mathematical planning models are closely related to
and rely upon traditional or nonmathematical planning. This
means, among other things, that the values of the exogenous
variables and parameters and also certain upper and/or lower
target values for some of the endogenous variables would not be
derived directly from statistical observations, but would be
based on figures given by traditional planners. (This is not to
say, however, that more or less sophisticated statistical estimation techniques would not be combined with experts' wguesstimates"
in traditional planning.) Finally, planning modelers in socialist
countries tend to concentrate more on the problems of how to fit
their models into the actual process of planning and make them
practically applicable and useful. Therefore, applied planning
models tend to be both theoretically and methodologically simpler
than those in the development planning literature.
Table 1 summarizes the major features of the two modeling
approaches. The list is far from complete and also it includes
afewconflicting or alien features. Thus, the question as to
whether models of the computable general equilibrium type could
be used in more or less the same function in planning as the
linear programming ones, is not trivial.
We do not have enough time or place here to go into details
(for such, seereferences 16 and 17) but the answer is affirmative. Certain techniques and certain types of models can be
viewed as natural extentions of the linear planning techniques
developed to date. Their study and adaption appear to open new
paths for central planning modeling practice. Before highlighting
Table 1.
MAJOR FEATURES OF APPLIED GENERAL EQULLIBRIUM (AGEM) AND OPTIMAL PLANNING (OPM) MODELS
Aspect
--
-
AGEM
OPM
-
Base of comparison
observed state (counterfactual simulation)
provisional plan (counterplan simulation)
Characteristic types
of variables
real, price, cost,
financial
mainly real, some
financial assets
Functional relationships based on
neoclassical theory
(e.g. production functions,
demand functions)
pragmatic considerations
(e.g. fixed norms,
structures)
Data basis
statistics (ex post)
plan information (ex ante)
Parameter estimation
techniques
direct and indirect
econometric estimation
mixed methods, heavy reliance on experts of various
fields
Decision criteria
individual profit and
utility maximization
overall consistency and
efficiency
S ~ e c i a lallocational
limits reflected by
varying rates of return
requirements (indirect)
individual bounds on variables (direct)
Mathematical form
nonlinear equation system,
locally unique solutions
(assumed)
linear inequalities with
alternative overall objective functions.
the possible advantages of such adaption, we would like to
illustrate how one can reinterpret the neoclassical forms
and adapt them to planning models.
3.
ILLUSTRATION: FOREIGN TRADE IN THE TWO VERSIONS OF
~~CROECONOMIC
MODELS
With the following simple example we try to facilitate
our discussion and the comparison of programming and general
equTlibrium approach. We will concentrate our attention on the
treatment of export and import in different multisectoral models.
For the sake of simplicity we will use an extremely stylized,
textbook type of a model. We will assume that there is only
one sector whose net output (!?) is given (determined by available resources]. The only allocation problem is to divide !?
into domestic use (Cd) and export (Z). Export will be exchanged
for an imported commodity which is assumed to be a perfect substitute for home commodity. Intermediate use will be neglected.
Following the traditional linear programming approach,
and import (FM) prices will be treated as (exoexport (B
E
genously given) parameters of the model. ~ntroducingM for
import purchased and Cm for import used, our optional resource
allocation problem can be formulated in the following simple
way.
C
=
Cd + Cm
+
max
The solution of the above problem depends clearly on the
relation of PE and PM, i.e., on the terms of trade. The problem
*
of overspecialization appears here in a very clear way. If the
terms of trade are favorable ( > IM ) then everything will be
exported (Z = P) and only imported goods consumed (Cd = 0,
In case of unfavourable terms of trade,
= Pi = PE Z/FM).
m'
the optimal policy will be that of autarchy.
Let us assume for a moment that the terms of trade are
favorable at prices FE and Ply. The model builders will be aware
of the fact that PE is only an approximate value of the unit
export price, and that at such a price the export markets could
not absorb more than, say, Z amount of export. Adding 2 to
the model as an individual upper bound on Z would prevent it
producing a completely overspecialized solution. Z would be
clearly binding and the solution would be
-
It is also easy to see that the optimal values of the dual variables will be
-
where t is the shadow price of the individual bound, Z
.
The analysis of this hypothetical planning model would not
stop here, for we know that Z is a constraint on export at given
PE export prices. What if we changed FE, would 2 also change?
Suppose that, at least within certain limits, the answer is yes,
i.e., a decrease in the export price (BE) would increase the
In other words, the modeled
export absorbtive capacity )
economy faces less than perfectly elastic export demand. Let
D(PE ) be the export demand function. Instead of the rigid,
fixed export bound ( Z ) we could thus use the following f l e x i b l e
constraint:
* See, for example, reference 14 on the problem of overspecialization and on the use of individual bounds in macroeconomic
models.
treating at the same time PE as a variable in the balance of
payment constraint.
As is known, one will usually find in linear programming
models of nationwide resource allocation individual bounds in
import as well. Typically, it is the ratio of import to the
domestic source (m) which is forced into some bounds. In our
case
+
was not constrained. Let us introduce m and m- as upper and
lower bounds on m. In such a case our previous programming
model will have to be augmented by two additional constraints.
These might be written together as
+
Let ti and tm denote the corresponding new shadow prices. As
a result of the above modifications in the primal problem the
dual constraints corresponding to Cd and Cm will have to be
modified in the following way
The computable models of general equilibrium usually
follow a different approach. There, the dependence of the import share (m) is usually an explicit and continuous, smooth
function of the relative prices of the domestic and imported
commodities. In most case, constant elasticity functions are
used, such as
In the linear programming case observe that if the lower
limit on import is binding (neglecting degenerated solutions),
then we will have ti > 0 and Pd < 1 , Pm > 1. If the upper limit
is binding then ti > 0 and Pd < 1 , Pm
1. Otherwise Pm = Pd.
Reversing the argument we could say the following. If the
shadow price of the domestic commodity is less than that of
the imported commodity, then we will not import more than the
minimum required. In the opposite case, we will import as much
as possible. Otherwise the import volume will be determined by
other considerations. Formally
Thus, the import share can be treated formally as a function
of relative prices in this case too. The function in this case
is, however, not a smooth one. (See Figure 1.)
It is worth noting here that essentially the same restrictions on import could not have been implicitly achieved by
introducing a piecewise linear objective function. Such an
objective function in a planning model could be viewed as the
planners preference (utility) function with respect to the
domestic-import composition of available sources. (See Figure 2 . )
We would like to emphasize that the difference between the
import restriction modes in the case of linear programming and
computable equilibrium models can again be seen as the one
between f i x e d (rigid) and f l e x i b l e individual bounds. The
relative (shadow or equilibrium) price dependent import share
implies a variable (flexible) individual bound on import: the
larger the gap between the domestic and import commodity (shadow)
prices, the large the deviation from the observed (or planned)
import ratio (mo). In fact, allowing for a smooth variation of
the import share around its planned level in a plan coordination
model makes at least as much sense, if not more, that the usual
rigid restrictions.
A
computable
general
equilibrium
I
+
m -
-w
mO -
linear
programing
--
m
'dIPm
F i g u r e 1.
I m p o r t Share F u n c t i o n s
Figure 2 .
I m p o r t r e s t r i c t i o n built i n t o t h e o b j e c t i v e
function
We complete our example by replacing fixed bounds with
flexible ones. Suppose we have a linear programming model with
fixed individual bounds, both on export and on import shares.
Let us repeat the model in full here
C = Cm
+ Cd
+
max
If we want to replace the fixed individual bounds by
flexible ones, in the manner described and discussed earlier,
we can proceed in the following way. We can rewrite the above
linear model into a nonlinear one by replacing the objective
function with one reflecting import limitations and introducing
an export demand function as before. These replacements will
yield the following model (using constant elasticity forms):
c
=
(hm
cI;lQ
-1 /Q
+hd
cd- ) ~
+
max
For lack of place we cannot show here how the parameters
(the parameters
hm' hd and q can be determined from mo and p
of the import share function) and v i c e v e r s a . Parameter D in
the foreign trade balance is a constant term derived by
solving the following export demand function for PE
In case of a "real" model it might be difficult to handle
nonlinearities. In such cases, piecewise linear approximations
could save the linear character of the model. (See reference 9
for more details of such a approach). We want to turn the
reader's attention to an alternative approach.
With reasonable values for the parameters, an interior
solution to the programming problem can be expected. By interpreting Pd, Pm and V as Lagrangian multipliers for the
corresponding constraints, the dual part of the first order neccessary (~uhn-~ucker)
conditions for a maximum can be stated as
follows :
I
ac
(1
We can show that conditions (1) and (2) will, in fact, yield
the import share function
It is also fairly easy to see that we can replace the above programming model by the following simultaneous equations system
This latter form is almost identical with a typical computable general equilibrium model specification. The only
difference from a competitive equilibrium model is in equation
(14). In the latter, we would only have Pd = V P E . The difference can be viewed as that between a p l a n n e r s o p t i m u m and a
laissez-faire equilibrium. (For details see reference 17).
We close this subsection with a brief discussion on the
derived equation system. Counting the variables (m, Cd, Cmt M,
Z, Pm, Pdt PEt V J we find that there is one more variable than
equations. This might lead to overdetermination problems.
However, observe that all the equations are homogenous of degree
zero in variables Pm' Pd and V. Thus the level of all of them
can be set freely.
The solution of a general equilibrium equation system needs
special algorithms. These will be discussed in a separate paper
by A. Por (see reference 12). We want to mention here only our
experiences with a model containing 19 sectors and close to 500
variables. To get a solution needs on average less than a minute
on an ICL System 4/70 computer.*
4.
CONCLUSIONS
Both here and in the earlier papers we have shown that a
certain class of multisectoral general equilibrium models, by
proper reinterpretation of their elements, can be adopted to
support planning in socialist countries. We have also demonstrated how certain nonlinear formulations of substitution
possibilities could be utilized in macroprogramming models in
order to keep the model relatively small and generate
meaningful dual solutions.
One major advantage of the equilibrium framework is that
it makes the dual side of the model less distorted while explicitly taking into account the interaction of real and value
*
-
For a more detailed description of the model and its
solution algorithm, consult reference 13.
variables. Thus, it may help planning modelers to achieve a
better linkage between plans for real and value processes.
These two main planning functions are usually quite separate
from each other in both traditional planning and modeling.
Changes in relative prices, costs, tariffs, etc., are not
properly reflected in physical allocation models, while the
effects of production, import/export, and consumption decisions
are not always taken into consideration in price planning models.
The mixed, primal-dual formulation of the resource allocation problem requires and also make it possible to reinterpret
the notion of efficiency (shadow) prices. On the one hand, the
mixed form allows the model builder to explicitly introduce
shadow-price dependent resource allocation decisions into his
model. In our simple model, it was quite easy to see how the
efficiency-price-dependent foreign trade decisions related to
the programming problem formulation. In more complex models,
such price-dependent (mixed primal-dual) decision rules can be
used in describing consumption and resource use alternatives,
etc. (see reference 16). The qeneral eauilibrium (mixed primaldual) formulation allows also for combining econometrically
estimated, price dependent macrofunctions with the optimal
resource allocation approach.
On the other hand, the equilibrium formulation makes it
possible to incorporate price-formation rules that reflect
the actual process more accurately than the shadow prices of
(linear) programming models. For example, even with constant
returns to scale, it is possible to define prices that do
contain profits (mark-up). One can also take into account
changes in taxes and tariffs and see how these would affect
the allocation decisions.
These comments suggest that the possible use of general
equilibrium models is manifold and not limited to coordinating
a plan. In fact, we believe that these models could also be
used for either e x p o s t or e x a n t e simulation of various issues
of concern to planners. Using statistical estimates of the
model parameters, structurally similar models (especially their
multiperiod extensions) could be tested in the forecasting
phase of planning.
In the central planning context, it seems
promising to combine such models with reference path optimization
techniques (see reference 15). Research in this direction is
currently underway both at the International Institute for
Applied Systems Analysis and in the Hungarian Planning Office.
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